Metrics of curves in shape optimization and analysis - Andrea Carlo ...

Definition 8.3 (Flat curves) Let Z be the set of all α ∈ L 2 ([0, 2π]) such thatα(s) = a + k(s)π where k(s) ∈ Z and k is measurable, a = 2π − ∫ k ∈ lR, and|{k(s) = 0 mod 2}| = |{k(s) = 1 mod 2}| = π• Z is closed (by thm. 4.9 in Brezis [4]).• S \ Z contains the (representation α by continuous lifting of) all smoothimmersed curves.• Z contains the (representations α of) flat curves ξ, that is, curves ξ whoseimage is contained in a line.Example 8.4 An example of a flat curve is{s/ √ {2 s ∈ [0, π]ξ 1 (s) = ξ 2 (s) =(2π − s)/ √ 2 s ∈ (π, 2π] , α = π/2 s ∈ [0, π]3π/2 s ∈ (π, 2π]Proposition 8.5 S \ Z is a smooth immersed submanifold of codimension 3 inL 2 .Proof. By the implicit function theorem. Indeed, suppose by contradictionthat ∇Φ 1 , ∇Φ 2 , ∇Φ 3 are linearly dependant at α ∈ S, that is, there existsa ∈ lR 3 , a ≠ 0 s.t.a 1 cos(α(s)) + a 2 sin(α(s)) + a 3 = 0for almost all s; then, by integrating, a 3 = 0, therefore a 1 cos(α(s))+a 2 sin(α(s)) =0 that means that α ∈ Z.The manifold S \ Z inherits a Riemannian structure, induced by the scalarproduct of L 2 ; (critical) geodesics may be prolonged smoothly as long as theydo not meet Z.Even if S may not be a manifold at Z, we may define the geodesic distanced g (x, y) in S as the infimum of the length of Lipschitz paths γ : [0, 1] → L 2 goingfrom x to y and whose image is contained in S; 12 since d g (x, y) ≥ ‖x − y‖ L 2,and S is closed in L 2 , then the metric space (S, d g ) is metrically complete.We don’t know if (S, d g ) admits minimal geodesics, or if it falls in the samecategory as the Atkin example 3.34.8.1.1 Multiple measurable representationsWe may represent any Lipschitz closed arc parameterized curve ξ using a measurableα ∈ S.Definition 8.6 A measurable lifting is a measurable function α : lR → lRsatisfying (8.1). Consequently, a measurable representation is a measurablelifting α satisfying conditions Φ(α) = (2π 2 , 0, 0) (from Definition 8.2).12 It seems that S is Lipschitz-arc-connected, so d g (x, y) < ∞; but we did not carry on adetailed proof54